In this website, RSJ refers to a rotating frame of reference in which both the Sun and Jupiter are at rest. Such a frame cannot actually exist, because Jupiter’s orbit is slightly eccentric, so that in an attempted RSJ frame, Jupiter would oscillate toward and away from the Sun. Special features such as the Lagrangian points would also move. In this website, Jupiter’s orbit is changed to one with zero eccentricity, which greatly simplifies calculations, while retaining qualitative features fairly well. The RSJ frame is non-inertial, sometimes resulting in apparently bizarre trajectories, but so much can be learned from using it that it is frequently employed. The shapes of the trajectories are due to the fictitious forces that always occur in non-inertial frames. The shapes of the trajectories are due to the fictitious forces that always occur in non-inertial frames. Three examples are shown below. All were attempts to generate Trojans at the L4 and L5 points, but the positions and velocities didn’t result in stable orbits. Two objects escaped, and the third crashed into Jupiter. The large yellow blob in the centre represents the sun, while the small white one on the right represents Jupiter, both obviously not to scale.
All these examples had similar starting positions and starting velocities. They demonstrate that this particular situation is one in which small initial differences can result in very different consequences, even though the basic laws are very simple. In fact, some possible situations may be described as chaotic, in the strict mathematical sense of the term.